Applications of Derivatives

Formulas are organized based on their importance in the CAT exam.
1 Vertex Form of a Quadratic
2 Monotonicity of a Quotient
3 Tangency of a Curve to the x-Axis
4 Chain Rule for Related Rates
5 Derivative of an Inverse Relation
6 Monotonicity of Functions
7 Maximum Area of a Rectangle Under a Parabola
8 Second Derivative Test
9 Strict Monotonicity Using Derivative
10 Error in Second-Order Approximation
11 Derivative of a Function with Absolute Value
12 Darboux Property of Derivatives
13 Critical Point
14 Optimization with Symmetry
15 Second Derivative Test for Maximum
16 Second Derivative in Parametric Form
17 Derivative as Slope of Tangent
18 Necessary Condition for Local Extrema
19 Derivative of a General Logarithmic Expression
20 Rolle's Theorem
21 Second Derivative in Implicit Form
22 Condition for No Distinct Stationary Points of a Cubic
23 Necessary Condition for Point of Inflection
24 Mean Value Theorem (MVT)
25 Transformation of $$0^0$$
26 Cauchy's Mean Value Theorem
27 First Derivative and Slope Behaviour
28 Point of Inflection
29 Critical Points
30 Concavity
31 Slope of an Implicit Curve
32 Absolute Extrema on a Closed Interval
33 Second-Order Approximation
34 Derivative Criterion for a Constant Difference
35 Taylor's Theorem
36 Rolle's Theorem for Root Counting
37 Increasing Function from First Derivative
38 Rolle's Theorem for Polynomial Roots
39 First Derivative Test for Maxima/Minima
40 Derivative of an Implicit Function
41 Maximum Value of a Quadratic Function
42 Monotonicity of a Product
43 Concave Upward
44 Shortest Distance Between Two Curves
45 Decreasing Function
46 Angle θ with Positive x-Axis
47 Standard Binomial Expansion
48 Optimization by Second Derivative
49 Derivative of a Composite Cotangent
50 Second-Order Small-Change Approximation
51 Uniqueness of a Root Using Derivative
52 Parametric Slope
53 Derivative Criterion for Constant Product
54 Stationary Point
55 Existence of a Root
56 Second Derivative Test Failure
57 L'Hôpital's Rule at Infinity
58 Slope of a Curve
59 Function with Zero Derivative
60 Angle Between Two Curves
61 Maximum and Minimum of a Function with Parameter
62 Tangency of Two Curves
63 Derivative of a Polynomial Function
64 Maximum Product for Fixed Sum
65 Stationary Point Separation in a Cubic
66 Transformation of $$\infty^0$$
67 Maximum Number of Stationary Points of a Polynomial
68 Angle Between Two Curves
69 Darboux's Theorem for Derivatives
70 Derivative of a Function Defined by an Integral
71 Number of Solutions Using Monotonicity
72 Standard Maclaurin Expansion of Cosine
73 Vertical Tangent
74 Equality of Derivatives
75 Polynomial Root Bound Using Rolle's Theorem
76 First Derivative Sign Test
77 Horizontal Tangent in Parametric Form
78 Point of Inflection
79 Normal to a Curve
80 Repeated Differentiation and Polynomial Degree
81 Transformation of $$0\cdot\infty$$
82 Tangent to a Curve
83 Continuity as a Necessary Condition for Differentiability
84 Curve with Constant Slope
85 Derivative of the Inverse at a Corresponding Point
86 Area of a Rectangle Inscribed Under a Parabola
87 Maximum Value of a Product Using Logarithmic Differentiation
88 Logarithmic Form of a Variable Power
89 Derivative of an Inverse Function
90 Maximum Area of a Rectangle Inscribed in a Circle
91 Non-Decreasing Function
92 Differentiability of a Piecewise Function
93 Differential of a Composite Function
94 Little-o Relation
95 Number of Roots from the Derivative
96 Vertical Tangent in Parametric Form
97 Transformation of $$1^\infty$$
98 Minimum Distance Between a Point and a Curve
99 Optimization with a Constraint
100 Relative Rate of Change
101 Local Maximum
102 Stationary Point of Inflection
103 Repeated Application of L'Hôpital's Rule
104 Linearization
105 Logarithmic Differentiation for Powers
106 Transformation of $$\infty-\infty$$
107 Newton's Method
108 Tangency of a Line and a Curve
109 Existence and Uniqueness of a Solution
110 Absolute Minimum
111 Rolle's Theorem
112 Concavity
113 Condition for Minimum Distance
114 Taylor Polynomial
115 Error Approximation
116 Maximum or Minimum Value of a Quadratic
117 Relative Error
118 Decreasing Function
119 Maximum Product of Positive Numbers with Fixed Sum
120 Derivative of a Composite Inverse Cosine
121 Derivative of a Composite Secant
122 Decreasing Function from First Derivative
123 Monotonicity of a Composite Function
124 Tangent at a Point Using Gradient
125 Second Derivative Test for Minimum
126 Stationary Point of an Implicit Curve
127 Error in Linear Approximation
128 Leibniz Notation for Related Rates
129 Maximum and Minimum of a Function on an Open Interval
130 Linear Approximation
131 Lagrange Mean Value Theorem
132 Maximum and Minimum Distance
133 Inverse Function Criterion
134 Area of a Rectangle Inscribed in a Circle
135 Standard Maclaurin Expansion of Exponential
136 Maximum Area of a Rectangle with Fixed Perimeter
137 First-Order Small-Change Approximation
138 Derivative of a Composite Inverse Tangent
139 Monotonicity Criterion
140 Derivative Criterion for Constant Ratio
141 Horizontal Tangent
142 Normality of Two Curves
143 Local Approximation of a Function
144 Absolute Maximum
145 Local Minimum
146 Newton's Method Geometrical Interpretation
147 Derivative and Root Multiplicity
148 Root Uniqueness Criterion
149 Elasticity of a Function
150 Maclaurin Series
151 Third-Order Approximation
152 Increasing Function
153 AM-GM Equality Condition
154 Maximum of a Function from Its Derivative Sign
155 Repeated Root Condition
156 Approximation of a Root by First-Order Expansion
157 Lipschitz-Type Bound from Derivative
158 Monotonicity of a Ratio
159 Leibniz Rule for Variable Upper Limit
160 Parameter for Tangency
161 Instantaneous Rate of Change
162 Condition for Two Stationary Points of a Cubic
163 Standard Maclaurin Expansion of Natural Logarithm
164 Convexity Criterion
165 Maximum Error Estimate
166 Standard Maclaurin Expansion of Sine
167 Higher Derivative Test
168 Derivative of a Composite Cosecant
169 Normal at a Point Using Gradient
170 Mean Value Theorem for Derivative Bounds
171 Approximate Change in a Function
172 Nature of Quadratic Extremum
173 Related Rates
174 Orthogonal Intersection
175 Derivative of a Difference
176 Error in First-Order Approximation
177 Fermat's Theorem
178 Order of Smallness
179 Maximum Area of a Triangle with Fixed Base and Height Constraint
180 AM-GM Optimization Principle
181 Equation of Normal Line
182 Minimum of a Function from Its Derivative Sign
183 Derivative of a Rational Power
184 Minimum Sum for Fixed Product
185 Approximation Using Differentials
186 Equation of Tangent Line
187 Maximum Number of Turning Points
188 Concave Downward
189 Cauchy's Mean Value Theorem in Ratio Form
190 Taylor Polynomial Remainder Bound
191 Non-Increasing Function
192 L'Hôpital's Rule
193 Average Rate of Change
194 Maximum Area Under a Fixed Perimeter
195 Percentage Error
196 Optimization of a Cubic Function
197 Derivative Criterion for One-to-One Function
198 Rate of Change
199 Minimum Surface Area of a Cylinder for Fixed Volume
200 Optimization Procedure
201 Taylor Remainder in Lagrange Form
202 Rolle's Theorem as a Special Case of LMVT
203 Candidates for Absolute Extrema
204 Minimum Sum of Positive Numbers with Fixed Product
205 Optimization Principle